Flows in porous media: mathematical and numerical aspects
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Transcript Flows in porous media: mathematical and numerical aspects
Flow in porous media:
physical, mathematical and
numerical aspects
13/04/201 - Stavanger-CFD Workshop
5
Peppino Terpolilli TFE-Pau
OUTLINE
• Darcy law
• Mathematical issues
• Some models: Black-oil, Dead-oil,BuckleyLeverett………
• Numerical approach
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Darcy law
• Navier-Stokes equations:
v
1
vv p v f
t
• Darcy law:
q
• K
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K
p
is the matrix of permeability: porous
media characteristic
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Darcy law
• Continuum mechanics:
at a REV located at x :
( x)
K ( x)
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porosity: ratio of void to bulk volume
permeability: Darcy law
REV
4
Darcy law
• Darcy law:
empirical law (Darcy in 1856)
• theoretical derivation:
Scheidegger, King Hubbert, Matheron
(heuristic)
Tartar (homogeneization theory)
Stokes
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Darcy law
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Darcy law
• Poiseuille flow in a tube:
single-phase, horizontal flow
steady and laminar
no entrance and exit effects
R 2 p
v
8 L
v
R
L
p
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mean velocity
radius
length
pressure gradient
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Darcy law
• Poiseuille flow in a tube:
K
R2
8
unit: darcy m2 1012 m2
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Darcy law
• Different scale:
pore level: Stokes equations
lab: measures
numerical cell: upscaling
field: heterogeneity
Darcy law
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Darcy law
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Black-oil model
• Extended Darcy law:
qp
Kkrp
p
( p p p gD)
• krp relative permeability of phase p
• D the depth
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Darcy law
• Continuum mechanics:
at a REV located at x :
So , w, g
kr (S )
pc (S )
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saturation: fraction of pore volume
relative permeability
capillary pressure
REV
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Kr-pc
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Kr-pc
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Math issues
• For single-phase flows Darcy law leads to
linear equation:
C
p
div ( K ( x).p) f
t
• For multi-phase flow we recover nonlinear
equtions: hyperbolic, degenerate parabolic
etc…..
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Math issues
• The mathematical model is a system of PDE
with appropriate initial and boundary
conditions
• the coefficients of the equations are poorly
known stochastic approach
• geology + stochastic = geostatistic
K ( x, )
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A field….
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Math issues
Data:
• wells
: core, well-logging, well test
• extension: geophysic, geology
• scale problems and uncertainty
(geostatistic)
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Uncertainty
• SPDE:
p
div ( K ( x, ).p ) f
t
• These problems are difficult:
experimental design approach
‘ Grand projet incertitude ’
Industrial tools
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Black-oil model
• Hypotesis:
three phases: 2 hydrocarbon phases and
water
hydrocarbon system: 2 components
a non-volatile oil
a volatile gas soluble in the oil phase
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Black-oil model
• Hypotesis:
components
oil
oil
gas
water
oil
gas
gas
water
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phases
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Black-oil model
phases:
water: wetting
oil
: partially wetting
gas : non wetting
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Sw
So
Sg
saturation
saturation
saturation
Black-oil model
• Validity of the hypothesis:
dry gas
depletion, immiscible water or gas injection
oil with small volatility
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Black-oil model
• PVT behaviour: formation volume factor
Bo
Vo Vdg RC
Vo STC
;
Vg RC
Bg
Vg STC
;
VW RC
Bw
VW STC
• where:
volume of a fixed mass at reservoir
VRC
conditions
VSTC
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volume of a fixed mass at stock tank
conditions
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Black-oil model
• Mass transfer between oil and gas phases:
Vdg
RS
V
o STC
Vdg : gas component in the oil phase
Vo : oil component in the oil phase
functions of the oil phase pressure
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Black-oil model
• Thermo functions for oil:
160
140
120
1,35
1,3
1,1
1,25
1
1,2
0,9
1,15
0,8
0,7
1,1
1,05
0
100
200
300
400
P
Bo
0,6
muo
0,5
1
0
100
200
P (bars)
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300
0,4
400
muo (cP)
100
80
60
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1,4
1,2
20
0
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1,4
1,3
Bo (Sm3/m3)
Rs(m3/m3)
200
180
Black-oil model
• Mass balance:
water
oil
gas
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(S w w ) div( w qw ) Qw
t
(So o ) div ( o qo ) Qo
t
(S g g So dg ) div ( g q g dg qo ) Qg Qdg
t
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Black-oil model
• Extended Darcy law:
qp
Kkrp
p
( p p p gD)
• krp relative permeability of phase p
• D the depth
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Black-oil model
• Water:
• oil:
• gaz:
Kkrw
Sw
div
Pw
wgZ
0
t Bw
Bw w
Kkro
So
div
Po
ogZ
0
t Bo
Bo o
Kkrg
Sg
So
Rs
Pg ggZ
div
t Bg
Bo
Bg g
KkroRs
div
Po ogZ 0
Bo o
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Black-oil model
• saturation:
So Sw Sg 1
• capillary pressures:
pw po pcow
p g po pcog
• we obtain 3 equations with 3 unknowns:
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po , Sw , Sg
if
po pb
po , Sw , Rs
if
po pb
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Black-oil model:boundary conditions
• Boundaries
closed: no flux at the extreme cells
aquifer: source term in corresponding cells
• wells:
Dirichlet condition: bottom pressure
imposed
Neumann condition: production rate
imposed
source terms for perforated cells (PI)
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Black-oil model: initial conditions
• capillary and gravity equilibrium
• pressure imposed in oil zone at a given depth
• oil pressure in all cells and then Pc curves
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Black-oil model: theoretical results
• Antonsev, Chavent, Gagneux:
existence results for weak solutions
• PME: porous media equation
more resuts: Barenblatt, Zeldovich,
Benedetti,…Vazquez.
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